Erdős Problem #710

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

For integers $a,b$, $a\mid b$ means $b=ac$ for an integer $c$. Coprimality is $\gcd(a,b)=1$, equivalently absence of a common prime divisor. All factorization variables in this entry are positive unless explicitly stated otherwise. [S1] [S2]

Statement recorded in the supplied source.

Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Obtain an asymptotic formula for $f(n)$. [S1]

Short English statement

How long an interval just above an integer is needed to assign a different multiple to each positive integer up to that integer?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2273 (EP-710), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 710, https://www.erdosproblems.com/710. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 P. Erdős and C. Pomerance, Matching the natural numbers up to n with distinct multiples in another interval, Indagationes Mathematicae 42 (1980), 147-161. https://math.dartmouth.edu/~carlp/PDF/matching.pdf. Bibliographic information from the supplied record.

• S4 Erdös, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. Bibliographic information from the supplied record.